## Well-Posedness and Convergence Rates of Three-Dimensional Incompressible Euler Flows in Axisymmetric Nozzles with Symmetric Body

Lin Jie,, Wang Tianyi,*

Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan 430070

 基金资助: 国家自然科学基金.  11971307

 Fund supported: The NSFC.  11971307

Abstract

This paper studies the three-dimensional incompressible Euler flows in axisymmetric nozzles with a symmetric body. The well-posedness is established by stream function method and barrier function. Base on the above well-posedness, the far field convergence rates of the solutions are studied: if the infinite nozzles are the flat boundary outside the finite length, the solution of the equation converges to an asymptotic state at the exponential rate; if the infinite nozzles converge to the flat boundary with the polynomial rates, the solutions converge to the asymptotic states at the same polynomial rates.

Keywords： Incompressible Euler equations ; Axisymmetric Nozzles ; Well-posedness ; Convergence rates

Lin Jie, Wang Tianyi. Well-Posedness and Convergence Rates of Three-Dimensional Incompressible Euler Flows in Axisymmetric Nozzles with Symmetric Body. Acta Mathematica Scientia[J], 2023, 43(1): 219-237 doi:

## 1 引言

$$$\left\{\begin{array}{ll} (\rho u_{1})_{x_{1}}+(\rho u_{2})_{x_{2}}+ (\rho u_{3})_{x_{3}}=0,\\ (\rho u_{1}u_{1})_{x_{1}}+(\rho u_{1}u_{2})_{x_{2}}+ (\rho u_{1}u_{3})_{x_{3}}+p_{x_{1}}=0,\\ (\rho u_{1}u_{2})_{x_{1}}+(\rho u_{2}u_{2})_{x_{2}}+ (\rho u_{2}u_{3})_{x_{3}}+p_{x_{2}}=0,\\ (\rho u_{1}u_{3})_{x_{1}}+(\rho u_{2}u_{3})_{x_{2}}+ (\rho u_{3}u_{3})_{x_{3}}+p_{x_{3}}=0,\\ (u_{1})_{x_{1}}+( u_{2})_{x_{2}}+ ( u_{3})_{x_{3}}=0, \end{array}\right.\label{1}$$$

${f_2}(x) = \left\{ \begin{array}{ll} {\tilde f_2}(x),\quad& x \in \left[ {L_1,L_2} \right],\\ 0,\quad& x \in \Bbb R \setminus [L_{1},L_{2}]. \end{array} \right.$

$$$(u_{1},u_{2},u_{3})\cdot\overrightarrow{n}=0, \quad(x_1,x_2,x_3)\in \partial \Omega\backslash\Pi, \label{1.12}$$$

$u_{1}^{2}+u_{2}^{2}+u_{3}^{2}=U^{2}+V^{2}+W^{2}.$

$B=\frac{U^{2}+V^{2}+W^{2}}{2}+\frac{p}{\rho}.$

$$$\left\{\begin{array}{ll} (r\rho U)_{x}+(r\rho V)_{r}=0,\\ (r\rho U^{2})_{x}+(r\rho UV)_{r}+rp_{x}=0,\\ (r\rho UV)_{x}+(r\rho V^{2})_{r}+r p_{r}=\rho W^{2},\\ (r\rho U(rW))_{x}+(r\rho V(rW))_{r}=0,\\ (rU)_{x}+(rV)_{r}=0. \end{array}\right.\label{1.19}$$$

$(x,r)-$ 坐标系下, 管道的边界为

$\Gamma_{1}=\{(x,r)|x\in(-\infty,+\infty),r=f_2(x)\}$

$\ \Gamma_{2}=\{(x,r)|x\in(-\infty,+\infty),r=f_1(x)\}.$

$$$(U,V,W)\cdot\overrightarrow{n}=0,\qquad (x,r)\in \Gamma_{1}\backslash\Pi,\label{1.14b}$$$

### 2.2 主要定理

$$$\label{AC2.14} U'_-(0)=\rho'_-(0)=W_-(0)=0,$$$
$$$\label{AC2.15} \rho_-(r)>0,\quad U_-(r)>0,\quad r\in\left(0,1\right),$$$
$$$\label{MT2.15} \left(\left(U^2_-+W^2_-\right)\rho \right)'(r)>0,\quad \delta r^2\left(\left(U^2_-+W^2_-\right)\rho_- \right)'(r)\geq \left(r^2W^2_-\rho_- \right)'(r), \quad r\in\left(0,1\right),$$$
$$$\label{MT2.16} \bigg(\frac{(U_-^2+W_-^2)'+\frac{2W_-^2}{r}}{rU_-}\bigg)'(r)\geq0,\qquad r\in[0,1],$$$
$$$\label{MT2.17} \bigg(\frac{(r^2W_-^2)'}{rU_-}\bigg)'(r)\leq0, \quad r\in[0,1],$$$
$$$\label{T2.17} \|\rho'_{-}\|_{C^{1,1}}\leq \delta,$$$

$$$\label{ping} f_1(x)=1, \quad x\le-K,$$$

$$$\Vert{U-U_-}\Vert_{C^{\alpha}(\Omega_{x-1,x+1})}+\Vert{V}\Vert_{C^{\alpha}(\Omega_{x-1,x+1})}\le C|x|^{-l},\quad x \le -K^{'},$$$

### 2.3 负无穷远处的状态

$$$\label{1.24} \rho\rightarrow \rho_-(r),$$$
$$$\label{1.25} U\rightarrow U_-(r),$$$
$$$\frac{1}{2}(U^2+V^2+W^2)+\frac{p}{\rho}\rightarrow B_-(r)$$$

$$$\label{1.26x} W\rightarrow W_-(r).$$$

$$$\label{mass-infty} B_-(r)= \frac{(U_{-}(r))^2}{2}+\frac{(W_{-}(r))^2}{2}+\frac{ p_{-}(r)}{\rho_-(r)}.$$$

$$$m=\int_0^1s U_{-}(s){\rm d}s.$$$

### 2.4 流线守恒量和流函数的方程

$$$\left\{\begin{array}{ll} (rU)_{x}+(rV)_{r}=0,\\ (r\rho U)_{x}+(r\rho V)_{r}=0,\\ (rU(rW))_{x}+(rV(rW))_{r}=0,\\ (rUB)_{x}+(rVB)_{r}=0. \label{LDF} \end{array}\right.$$$

$$$\label{equ:3.8xx} \psi_-(r)=\int_0^r sU_{-}(s) ds.$$$

$\kappa(m):=\psi^{-1}_{-}(m) $$\left[m\right]$$\left[1\right]$ 的一一映射. 直接计算得到

$$$\kappa'(\psi_-)=\frac{1}{\kappa U_{-}(\kappa)} \quad\mbox{和}\quad \kappa''(\psi_-)=\left(-\frac{1}{\kappa(\psi_-)}-\frac{U_{-}'(\kappa)}{U_{-}(\kappa)}\right)(\kappa')^2(\psi_-).$$$

$$$\label{E3.13} \Delta\psi+b_{i}\partial_{i}\psi={\cal F}(\psi,r),$$$

$\begin{eqnarray*} &&b_1(\psi,r)=-\frac{\partial_1\psi{\cal G}'}{2{\cal G}},\\ &&b_2(\psi,r)=-\frac{\partial_2\psi{\cal G}'}{2{\cal G}}-\frac{1}{r},\\ &&{\cal F}(\psi,r)=r^2{\cal G}\left(\frac{{\cal B}}{{\cal G}}\right)'-\frac{{\cal G}}{2}\left(\frac{{\cal W}^2}{{\cal G}}\right)'. \end{eqnarray*}$

### 3 定理1.1的证明

${\cal G}(\psi)$, ${\cal B}(\psi)$${\cal W}(\psi)$$[m]$ 延拓到 $\Bbb R$.

$$$\dot{{\cal G}}(s)=\left\{ \begin{array}{ll} {\cal G}'(s),& \mbox{当}\ 0\leq s\leq m,\\ {\cal G}'(m)\frac{2m-s}{m},&\mbox{当}\ m\leq s\leq 2m,\\ 0,& \mbox{当}\ s\geq2m\ \mbox{或}\ s\leq 0, \end{array}\right.$$$
$$$\label{4.3} \dot{{\cal B}}(s)=\left\{ \begin{array}{ll} \left(\frac{{\cal B}}{{\cal G}}\right)'(s),& \mbox{当}\ 0\leq s\leq m,\\ \left(\frac{{\cal B}}{{\cal G}}\right)'(m)\frac{2m-s}{m},&\mbox{当}\ m\leq s\leq 2m,\\ 0, & \mbox{当}\ s\geq2m\ \mbox{或}\ s\leq-0 \end{array}\right.$$$

$$$\dot{{\cal W}^2}(s)=\left\{ \begin{array}{ll} \left(\frac{{\cal W}^2}{{\cal G}}\right)'(s),& \mbox{当}\ 0\leq s\leq m,\\ \left(\frac{{\cal W}^2}{{\cal G}}\right)'(m)\frac{2m-s}{m}, & \mbox{当}\ m\leq s\leq 2m,\\ 0,& \mbox{当}\ s\geq2m\ \mbox{或}\ s\leq 0. \end{array}\right.$$$

$\begin{matrix} \label{3.16} \tilde{{\cal G}}(s)&=&{\cal G}(0)+\int^{s}_{0}\dot{{\cal G}}(t){\rm d}t, \\ \tilde{{\cal B}}(s)&=&\tilde{{\cal G}}(s)\left(\frac{{\cal B}}{{\cal G}}(0)+\int^{s}_{0}\dot{{\cal B}}(t){\rm d}t\right),\\ \tilde{{\cal W}^2}(s)&=&\tilde{{\cal G}}(s)\left(\frac{{\cal W}^2}{{\cal G}}(0)+\int^{s}_{0}\dot{{\cal W}}^{2}(t){\rm d}t\right). \end{matrix}$

$\begin{eqnarray*} {\cal K}(0,s,r)&=&\left(\frac{\tilde{{\cal B}}}{\tilde{{\cal G}}}\right)'(s) -\frac{1}{2r^2}\left(\frac{\tilde{{\cal W}}^{2}}{\tilde{{\cal G}}}\right)'(s) \\ &=&\kappa'\left(\left(\frac{\left(U^2_-+W^2_-\right)\rho_-}{2}\right)'(\kappa)+\frac{\rho_{-}W_{-}^{2}}{\kappa}(\kappa) -\frac{1}{2r^2}\left(\rho_{-}W_-^{2}\right)'(\kappa)\right). \end{eqnarray*}$

$\begin{eqnarray*} &&\left(\frac{\left(U^2_-+W^2_-\right)}{2}\rho_-\right)'(\kappa)+\frac{\rho_{-}W_{-}^{2}}{\kappa}(\kappa) -\frac{1}{r^2}\left(\rho_{-}\frac{W_-^{2}}{2}\right)'(\kappa)\\ &\geq& \left(1-\delta\right)\left(\frac{\left(U^2_-+W^2_-\right)}{2}\rho_-\right)'(\kappa)+\frac{\rho_{-}W_{-}^{2}}{\kappa}(\kappa)\\ &\geq& \delta'>0. \end{eqnarray*}$

$\psi^{(k)}_L(P_{\mbox{max}})\geq \phi(P_{\mbox{max}}),\quad \nabla\psi^{(k)}_{L}=\nabla\phi= \left(0,\frac{2m}{b^2}(r+k)\right),$

$\begin{eqnarray*} \Delta\left(\psi^{(k)}_{L}-\phi\right)+b_{i}^{(k)}\partial_i\left(\psi^{(k)}_{L}-\phi\right) =r^2{\cal G}\left(\frac{{\cal B}}{{\cal G}}\right)'-\frac{{\cal G}}{2}\left(\frac{{\cal W}^{2}} {{\cal G}}\right)' -\frac{{\cal G}'}{2{\cal G}}\frac{4m^2}{b^4}(r+k)^2. \end{eqnarray*}$

$$$\label{C3x} \Delta(\psi^{(k)}_{L}-\phi)+b_{i}^{(k)}\left(\psi^{(k)}_{L},r+k\right)\partial_i(\psi^{(k)}_{L}-\phi)\geq0.$$$

$0\leq \psi^{(k)}_{L}\leq \frac{m}{b^2}(r+k)^2.\label{C3}$

$\left|\frac{\nabla\psi_{L}}{r}\right|\leq Cm,\qquad 0<r<\frac{b}{2}.$

$$$\label{equ:5.2} \left\{\begin{array}{ll} (r U)_{x}+(r V)_{r}=0,\\ U_r-V_x=0, \end{array}\right.$$$

$$$\label{irrotational} \left\{\begin{array}{ll} \mbox{div}_{(x, r)}\left(\frac{\nabla\bar{\psi}}{r}\right)=0, \quad &(x, r)\in \Omega_{L},\\ \bar{\psi}=\frac{mr^2}{f^{2}_{1}(x)},\ \quad &(x, r)\in \partial\Omega_{L}. \end{array}\right.$$$

$$$\frac{\partial\bar{\psi}_{L}}{\partial\eta}=-\partial_{1}\bar{\psi}_{L} f'_{2}(x)+\partial_{2}\bar{\psi}_{L}>0,$$$

$$$\partial_{1}\bar{\psi}_{L}+\partial_{2}\bar{\psi}_{L} f'_{2}(x)=0,$$$

$$$\frac{\partial\bar{\psi}_{L}}{\partial\eta}=\partial_{2}\bar{\psi}_{L}(1+f'_{2}(x)^{2})>0.$$$

$\begin{eqnarray*} \phi=\psi_L-\bar{\psi}_L\qquad \mbox{和}\qquad\psi_{\tau}=\bar{\psi}_L+\tau(\psi_L-\bar{\psi}_L),\quad \mbox{其中 }\ \tau\in(0,1). \end{eqnarray*}$

$$$\mbox{div}\left(\frac{\nabla\psi_L-\nabla\bar{\psi}_L}{r}\right)=r\left({\cal B}'-p\left({\cal G}\right)'\right)-\frac{{\cal W}{\cal W}'}{r}.$$$

$$$\Delta \phi-\frac{1}{r}\partial_2\phi=r^2\left({\cal B}'(\psi)-p\left({\cal G}\right)'(\psi)\right)-{\cal W}{\cal W}'.$$$

$\begin{matrix} \|\phi\|_{C^{2,\alpha}(K)}\leq C\|({\cal G}',{\cal B}',{\cal W}')\|_{C^{0,1}([m])} \leq C\delta.\label{7.16} \end{matrix}$

$\|\psi_L-\bar{\psi}_L\|_{C^{2, \alpha}(K)}\leq C\delta.$

$$$\left|\psi_L\right|\leq m+C\sup\limits_{\Omega_L}\tilde{{\cal F}},$$$

$$$-C|\sup\limits_{\Omega_L}\tilde{{\cal F}}|\leq\psi_L\leq m+C|\sup\limits_{\Omega_L}\tilde{{\cal F}}|.$$$

$$$[\psi_L]_{1,\mu;\Omega_L}\leq C\left(1+m+\|\tilde{{\cal F}}\|_{0;\Omega_L}\right)$$$

$$$\|\psi_L\|^{(-1-\alpha)}_{2,\alpha,\Omega_L}\leq C\left(\|\tilde{{\cal F}}\|_{0;\Omega_L}+\|\psi_L\|_{0;\Omega_L}\right), 0<\alpha<\mu.$$$

$$$\left\{\begin{array}{ll} \mbox{div}_{(x, r)}\left(\frac{\nabla \psi_1-\nabla \psi_2}{r}\right)={\cal K}\left(\left|\frac{\nabla\psi_1}{r}\right|^2,\psi_1,r\right)-{\cal K}\left(\left|\frac{\nabla\psi_2}{r}\right|^2,\psi_2,r\right),\quad &(x,r)\in \Omega,\\ \psi_1-\psi_2=0,\quad& (x,r)\in\partial\Omega. \end{array}\right.$$$

$\begin{eqnarray*} \phi=\psi_1-\psi_2\qquad \mbox{和}\qquad\psi_{\tau}=\psi_1+\tau(\psi_1-\psi_2). \end{eqnarray*}$

$$$\label{UN} \Delta \phi-\frac{1}{r}\phi_{r}-r\widehat c_i\partial_i\phi-r\widehat d\phi=0,$$$

$\begin{eqnarray*} \widehat c_{i}\left(\frac{\nabla\psi_\tau}{r},\psi_\tau,r\right) =\int_{0}^{1}\frac{{\cal G}'\partial_{i}\psi_\tau}{2{\cal G}r}{\rm d}\tau,\qquad \widehat d\left(\frac{\nabla\psi_\tau}{r},\psi_\tau\right)=\int_{0}^{1}\partial_{\psi_\tau}{\cal K}\left(\left|\frac{\nabla\psi_\tau}{r}\right|^{2},\psi_\tau,r\right){\rm d}\tau. \end{eqnarray*}$

$\eta\in C^{\infty}_0(\Bbb R)$ 满足: 当 $|s|<L$ 时, $\eta=1$; 当 $|s|>L+1$ 时, $\eta=0$. 将式(3.26)乘 $\eta^2(x)\phi$, 再在 $\Omega$ 上积分得

$\iint_{\Omega}\Delta \phi\eta^2(x)\phi {\rm d}x{\rm d}r=\iint_{\Omega}\left(\frac{1}{r}\phi \phi_{r}+r\widehat c_i\partial_{i}\phi\eta^2(x)\phi+r\widehat d\eta^2(x)\phi^2\right){\rm d}x{\rm d}r.$

$\begin{eqnarray*} &&\iint_{\Omega}\eta^2|\nabla \phi|^2{\rm d}x{\rm d}r\nonumber\\ &=&-2\iint_{\Omega}\phi\eta\nabla \eta \cdot\nabla\phi {\rm d}x{\rm d}r-\iint_{\Omega}\left(\frac{1}{r}\phi_{r}+r\widehat c_i\partial_{i}\phi\right)\eta^2(x)\phi {\rm d}x{\rm d}r-\iint_{\Omega}r\widehat d \eta^2(x)\phi^2{\rm d}x{\rm d}r.\nonumber \end{eqnarray*}$

$\begin{eqnarray*} d&=&r{\cal B}''-rp{\cal G}''-r\frac{{\cal B}'{\cal G}'-r^{-2}{\cal W}{\cal W}'{\cal G}'-p({\cal G}')^2}{{\cal G}}-\frac{({\cal W}')^2+{\cal W}{\cal W}''}{r}\nonumber\\ &=&r(B_{-}''(\kappa')^2+B_{-}'(\kappa''))\nonumber\\ &&-(\kappa^2 W_{-}^2)''(\kappa')^2+ r (\kappa^2 W_{-}^2)'(\kappa)''\nonumber\\ &&-(\kappa')^2\left(-r\rho_{-}'B_{-}'+r^{-1}(\kappa^2W_{-})'{\rho_{-}}'+\rho_{-}^{-3}p_-\left(\rho_{-}'\right)^2\right)\nonumber\\ &&-rp\left(\left(\frac{1}{\rho_{-}}\right)''(\kappa')^2+\left(\frac{1}{\rho_{-}}\right)'\kappa''\right)\nonumber\\ &:=&\sum\limits_{i=1}^{4}I_{i}.\nonumber \end{eqnarray*}$

$\begin{eqnarray*} &&{\cal B}'(\psi)=\kappa'B_-',\qquad {\cal B}''(\psi)=(\kappa')^2B_-''+\kappa''B_-';\nonumber\\ &&{\cal G}'(\psi)=\kappa'\rho_-',\qquad {\cal G}''(\psi)=(\kappa')^2\rho_-''+\kappa''\rho_-';\nonumber\\ & &{\cal W}'(\psi)=\kappa'W_-+\kappa\kappa'W_-',\qquad {\cal W}''(\psi)=\kappa(\kappa')^2W_-''+\kappa\kappa''W_-'+2(\kappa')^2W_-'+\kappa''W_-.\nonumber \end{eqnarray*}$

$\begin{eqnarray*} &&B_{-}''(\kappa)(\kappa')^2+B_{-}'\kappa''\nonumber\\ &=&(\kappa')^2\left(\left(-\frac{1}{\kappa}-\frac{U_{-}'}{U_-}\right)\left(\left(\frac{U_{-}^2+W_{-}^2}{2}\right)'+\frac{W_{-}^2}{\kappa}\right)(\kappa)+\frac{W_{-}^2}{\kappa}'+\left(\frac{U_{-}^2+W_{-}^2}{2}\right)''(\kappa)\right)\nonumber\\ \nonumber\\ &&+(\kappa')^2\left(p_{-}\left(\frac{1}{\rho_{-}}\right)'\right)'(\kappa)+\kappa''p_{-}\left(\frac{1}{\rho_{-}}\right)' \nonumber\\ &=&(\kappa')^2\kappa U_{-}(\kappa)\left(\frac{\left(\frac{U_{-}^2+W_{-}^2}{2}\right)'(\kappa)+\frac{W_{-}^2}{\kappa}}{\kappa U_{-}}\right)'+(\kappa')^2\left(p_-\left(\frac{1}{\rho_{-}}\right)'\right)'+\kappa(\psi_-)''p_-\left(\frac{1}{\rho_{-}}\right)'.\nonumber \end{eqnarray*}$

$$$B_{-}''(\kappa)(\kappa')^2+B_{-}'\kappa''\geq -\frac{\delta'}{4}.$$$

$$$\partial_x\bar{\psi}=\lim_{\Delta x\rightarrow0}\frac{\bar{\psi}\left(x+\Delta x\right)-\bar{\psi}\left(x\right)}{\Delta x}.$$$

$$$\bar{\psi}\left(x+\Delta x\right)=\bar{\psi}\left(x\right).$$$

$$$\rho\rightarrow \rho_{-}, \qquad U\rightarrow U_-, \qquad V\rightarrow 0, \qquad W\rightarrow W_-, \qquad p\rightarrow p_-.$$$

$\lim_{x\rightarrow+\infty}\psi(x,r)=\bar{\psi}_{+},$

$$$\rho\rightarrow \rho_{+}, \qquad U\rightarrow U_+, \qquad V\rightarrow 0, \qquad W\rightarrow W_+, \qquad p\rightarrow p_+.$$$

### 3.3 具有角点的情况

$$$w=\frac{C_1m}{R^{1+\beta}}s^{1+\beta}\sin(A\theta+\theta_*),$$$

$\Omega^{\varepsilon}\cap \partial B_R(x_0)$, $\psi^{\varepsilon}\leq m\leq w$ 和在 $B_R(x_0)\cap\partial\Omega_{\varepsilon}$, $\psi^{\varepsilon}=0\leq w$.$\Omega^{\varepsilon}\cap B_R(x_0)$ 内, 当 $\beta$ 足够小时

$\begin{eqnarray*} \Delta w&=&\frac{C_1m}{R^{1+\beta}}(-A^2+1+\beta)s^{\beta-1}\sin(A\theta+\theta_*)+\frac{C_1m}{R^{1+\beta}}\beta(\beta+1)s^{\beta-1}\sin(A\theta+\theta_*)\\ & \leq&-Cs^{\beta-1} \leq\Delta\psi^{\varepsilon}. \end{eqnarray*}$

$$$0 \leq\psi^{\varepsilon}\leq Cs^{1+\beta},$$$

$$$w=m-\frac{C_1m}{R^{1+\beta}}s^{1+\beta}\sin(A\theta+\theta_*),$$$

### 4 定理1.2的证明

$\psi_-$是问题(3.29)中$a_B$的解, 记

$\Psi=\psi-\psi_-,\quad \Psi_\tau=\psi+\tau\Psi, \quad \tau\in[0,1].$

$$$L_0\Psi=\Delta\Psi-\frac{1}{r}\partial_2\Psi-r{\cal C}_{i}\partial_i\Psi-r{\cal D}\Psi=0,$$$

$$${\cal C}_{i}=\widehat c_{i} \left(\frac{\nabla\Psi_\tau}{r},\Psi_\tau,r\right),$$$
$$${\cal D}=\widehat d\left(\frac{\nabla\Psi_\tau}{r},\Psi_\tau,r\right).$$$

$\eta(x,r)={\rm e}^{\mu r+\epsilon x}$, 其中 $\mu$$\epsilon 是正常数, 对 \eta 作用 L_0 \begin{matrix} L_{0}\eta&=&\partial_{11}\eta+\partial_{22}\eta-r{\cal C}_{1}\partial_{1}\eta-\left(\frac{1}{r}+r{\cal C}_{2}\right)\partial_{2}\eta-r{\cal D}\eta\nonumber\\ &=&{\rm e}^{\mu r+\epsilon x}\left(\epsilon^{2}+\mu\left(\mu-\frac{1}{r}\right)-\epsilon c_1-\mu r{\cal C}_{2}-r{\cal D}\right) \end{matrix} \mu 足够小, 有 \mu-\frac{1}{r}\leq -\frac{1}{2}.\epsilon 足够小, 有 \epsilon^{2}+\mu\left(\mu-\frac{1}{r}\right)\leq -\frac{\mu}{4}.{\cal D}>0, 令 \mu 足够小, 有 -\mu r{\cal C}_{2}-r{\cal D}\leq \frac{\mu}{8}. 于是, $$L_0\eta\leq -\frac{\mu}{8}{\rm e}^{\mu r+\epsilon x}.$$ \nu={\rm e}^{\mu r}, 同理得 $$L_0\nu\leq -\frac{\mu}{8}{\rm e}^{\mu r}.$$ K_{2} 是一个常数且 K_{2}>K_{1}, 存在一个常数 C_0 满足 $$|\Psi(K_2,r)|\le C_0 {\rm e}^{\epsilon K_2} {\rm e}^{\mu r}.$$ 对于任意固定的 \beta>0, 存在 s_0<K_1 满足 $$|\Psi(x,r)|\le \beta {\rm e}^{\mu r}, \quad x<s_0.$$ 注意到 $$\Psi(x,0)=\Psi(x,1)=0, \quad x<K_2.$$ 因此, 对于任意的 s<s_0, 满足 $$\left\{\begin{array}{ll} L_0\Psi=0>L_0\left(C_0\eta+\beta {\rm e}^{\mu r}\right), \quad& (x,r)\in\Omega_{s,-K_2},\\ \Psi\le C_0\eta+\beta {\rm e}^{\mu r},\quad &(x,r)\in\partial\Omega_{s,-K_2}. \end{array}\right.$$ 利用极大值原理得 $$\Psi\le C_0\eta+\beta {\rm e}^{\mu r}, \ \ (x,r)\in\Omega_{s,-K_2}.$$ s 的任意性得 $$\Psi\le C_0\eta+\beta {\rm e}^{\mu r}, \ \ (x,r)\in\Omega_{-\infty,-K_2}.$$ 相似地, 存在一个常数 C_1>0 满足 $$-\Psi\le C_1\eta+\beta {\rm e}^{\mu r}, \ \ (x,r)\in\Omega_{-\infty,-K_2}.$$ 于是, 对于任意的 \beta>0, $$|\Psi|\le C_2\eta+\beta {\rm e}^{\mu r}, \ \ (x,r)\in\Omega_{-\infty,-K_2}.$$ 这里的 C_2=\max \{ {C_0},{C_1}\} .\beta 的任意性得 $$|\Psi|\le C_2 {\rm e}^{\epsilon x},\ \ (x,r)\in\Omega_{-\infty,-K_2}.$$ 于是 $$\Vert\Psi\Vert_{C^{2,\alpha}(\Omega_{x-1,x+1})}\le C |\Psi|_{C^{0}(\Omega_{x-1,x+1})}\le C {\rm e}^{\epsilon x}, \ \ \ x\le-K_2,$$ 因此 $$\Vert U-U_0\Vert_{C^{1,\alpha}(\Omega_{x-1,x+1})}+\Vert V\Vert_{C^{1,\alpha}(\Omega_{x-1,x+})}\le C {\rm e}^{\epsilon x},\ \ \ x\le-K_2.$$ ### 4.2 边界以代数速率收敛到平直边界的情况 \Upsilon(x,r)=|-x|^{-l}{\rm e}^{\mu r}. 其中 \mu 是正常数. 直接计算得到 \begin{eqnarray*} \bar{L}\Upsilon&=&\partial_{11}\Upsilon+\partial_{22}\Upsilon-r{\cal C}_{1}\partial_{1}\Upsilon-\left(\frac{1}{r}+r{\cal C}_{2}\right)\partial_{2}\Upsilon-r{\cal D}\Upsilon\nonumber\\ &=&|x|^{-l}{\rm e}^{\mu r}\left(\frac{rl\left(l+1\right)}{x^{2}}+\mu\left(\mu-\frac{1}{r}\right)-\epsilon c_1-\mu r{\cal C}_{2}-r{\cal D}\right). \end{eqnarray*} \mu 足够小和 |x| 足够大得 $$\bar{L}\Upsilon(x,r)\leq -\frac{\mu}{8}|x|^{-l}{\rm e}^{\mu r}.$$ \hat{ f} =\min \{ {1},{f_2(x)}\}$$\hat{\Omega}=\{(x,r)| 0\le r \le \hat{f}\}$. 于是, 存在足够大的 $J_2>J_1$, 当 $x<-J_2$, 得

$$$|\Psi|=|\psi-\psi_0|\le|\psi-\frac{m_0}{2\pi}|+|\psi_0-\frac{m_0}{2\pi}|\le(\nabla\psi+\nabla\psi_0)|\hat{f}-1|\le C|x|^{-l}, \quad x\in\partial \hat{\Omega}.$$$

$J_3$ 是一个大于$J_2$ 的常数, 存在常数 $C$ 满足

$$$|\Psi(J_3, r)|\le C|J_3|^{-l}{\rm e}^{\mu r}.$$$

$$$|\Psi|\le \beta {\rm e}^{\mu r},\quad x<s_1.$$$

$$$\left\{\begin{array}{ll} L_0\Psi=0>L_0\left(C\upsilon+\beta {\rm e}^{\mu r}\right), \ \ &(x,r)\in \Omega_{s,-J_3},\\ \Psi\le C\upsilon+\beta {\rm e}^{\mu r}, \ \ &(x,r)\in \partial\Omega_{s,-J_3}. \end{array}\right.$$$

$$$|\Psi|\le C\upsilon+\beta {\rm e}^{\mu r}, \ \ (x,r)\in \Omega_{s,-J_3}.$$$

$\beta$$s$ 的任意性得

$$$|\Psi|\le C|x|^{-l}, \ \ (x,r)\in \Omega_{s,-J_3}.$$$

$$$\Vert\Psi\Vert_{C^{2,\alpha}(\Omega_{x-1,x+1})}\le C |\Psi|_{C^{0}(\Omega_{x-1,x+1})}\le C|x|^{-l},\ \ \ x\le-J_3.$$$

$$$\Vert U-U_-\Vert_{C^{1,\alpha}(\Omega_{x-1,x+1})}+\Vert V\Vert_{C^{1,\alpha}(\Omega_{x-1,x+1})}\le C|x|^{-l},\ \ \ \ x\le-J_3.$$$

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